a relation between structures and network flows through graph representation
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A relation between Structures and Network Flows Through Graph Representation. It was found that the same type of representation, a Graph , can be associated with more than one domain, say Network Flow and One-Dimensional Structures - PowerPoint PPT PresentationTRANSCRIPT
A relation between Structures and Network Flows Through Graph
Representation It was found that the same type of representation, a Graph,
can be associated with more than one domain, say Network Flow and One-Dimensional Structures
Then for each engineering system si’ Structure and siNetwork we can construct a system so that
T’(s’i) =si
Network
si
Ts’i
Structure
B
A
ts
1 4
23
5
P
1
4
2
3
5
P
B
Network Flow and its connection to structures
• The following slides will demonstrate the close relation between maximum flow in networks and applying a maximal force in a one dimensional structure.
One dimensional structures
• In 1965, William Prager established the relation between network flow and plastic theory for one dimensional structures.
• Prager used graph theory in order to establish the connection between the two seemingly different areas.
One dimensional structures
• A one-dimensional structure is a solid structure built of rods and discs, where all rods are parallel to each other.
Ground
Rods
Disc
P
Network Flow and its connection to structures
• The rules are pretty straight forward – each disc is replaced by a vertex and each rod by an edge.
structure network
A
B
ts
1 4
2
3
5
P
1
4
2
3
5
P
B
A
s
t
One dimensional structures
• Applying more than the allowable force will in turn transform the structure into a mechanism.
• Cable – A rod which preserves its length but cannot accept any compression (Recski)
• Strut – A rod which cannot accept any tension
P s
A B
t
Energy preservation law:
n
i
m
jFjpi ji
FP1 1
P
s
B
A
t
Network Flow - Example
3 5
1
10 5
A
B
st
Network Flow - Example
One can see that the maximum flow is 9.
3 51
10 5
A
B
st
One dimensional structures- and its link to networks
• Let’s assume now that we have the following one-dimensional structure:
p s
3 10 A
B 1 5 5
t
One dimensional structures- and its link to networks
• Where the numbers refer to the maximal allowable force in each rod.
p s
3 10 A
B 1 5 5
t
One dimensional structures- and its link to networks
• Now we have to find the maximal force that can be applied on the structure before any of its discs start moving.
p s
3 10 A
1
B t
5 5
One dimensional structures- and its link to networks
• The optimal solution here is 9, and one of the paths is marked in the graph:
A
B
s t
t
3 03 0
5 2 1
5 2 110 5 4
10 5 4
5 0
5 0
1 01 0
P
B
A
s
• Now, finding a cutset in the structure in which all rods are saturated means we have a mechanism:
Augmenting Path
Min Cut Max Flow
Isomorphic
Network Flow and its connection to structures
• What we get is the graph representation of the structure which is isomorphic to the network.
• Now, finding the maximal allowable force is the same objective as finding the maximum allowable flow.
structure Network
p
s
1 2 1 4
A 3
4 3 B 2 5
5 P
t
A
B
s t
A relation between Structures and Network Flows Through Graph
Representation It was found that the same type of representation, a Graph,
can be associated with more than one domain, say Network Flow and One-Dimensional Structures
Then for each engineering system si’ Structure and siNetwork we can construct a system so that
T’(s’i) =si
Network
si
Ts’i
Structure
B
A
ts
1 4
23
5
P
1
4
2
3
5
P
B
A relation between Structures and Linear Programming Through
Matroid theory It was found that the same type of representation, a
Matroid can be associated with more than one domain, say LP (Linear Programming) and Multi-Dimensional Structures
Then for each engineering system siLP we can construct a system si’ Structures so that
T(si)=mi =T’(s’i)Matroid
mi
LP
si
Ts’i
Structures
T’
n
iiiiXC
mibxa i
n
jjij ...1,
1
0ix
Max
st Q(M)*F=0B(M)*D=0
Frames
Trusses
Electronic circuits
Dynamical system
Static lever system
PGRPotential Graph Representation
LGRLine Graph
Representation
RGRResistance Graph
Representation
FGRFlow Graph
Representation
FLGRFlow Line Graph Representation
PLGRPotential Line Graph
Representation
Planetary gear systems
Determinate beams
Serial robots Stewart platformPillar
system
The map of domains
Dual
RMRResistance Matroid
Representation
Plane kinematical linkage
LPLinear Programming
Operational research
DualNetwork Flow
n
iiiiXC
mibxa i
n
jjij ...1,
1
0ix
Max
st
Q(G)*F=0
B(G)*D=0
F=k*D
Q(M)*F=0
B(M)*D=0
Matroid Representation
The scalar cutset matrix defines the matroid MQ=(S,F) where S is the set of columns of Q(G) and F is a family of all linearly independent subsets of S.
P
321S={1,2,3,P)
F={{1},{2},{3},{P},{1,2},{1,3},{1,P},{2,3},
{2,P},{3,P}}
Structures Network Flows Matroid
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Choosing determinate
structure
Choosing augmenting path
Choosing a base
Choosing a cutChoosing a cutRemoving these rods makes the structure not rigid
Choosing a self stress
Choosing a cycleChoosing a set of linearly dependant
members
Force law Flow law Q(M)*F=0
Allowable force in each rod
The weights in the edges
The maximum values of the
members
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The maximum values of the members
Choosing a base
B(M)*D=0
Q(M)*F=0
Choosing a set of linearly dependant members
A group of manufacturing
workers
A group of manufacturing workers and an administrative/non-manufacturing worker
The sum of hours that a
worker manufactures the
products equals to the sum
of his Working hours
A unit’s cost equals to the
sum of the hours multiplied
by the workers’ wages
The workers’ hour
constraint
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Matroid Operational Research
Multi-Dimensional Structures
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